Gerard ’t Hooft, quant-ph/0604008

Erice, September 6, 2006

Utrecht University

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Motivations for believing in a deterministic theory:

0. Religion1. “Quantum cosmology”2. Locality in General Relativity and String Theory3. Evidence from the Standard Model (admittedly weak).4. It can be done.

“Collapse of wave function” makes no sense QM for a closed, finite space-time makes no senseString theory only obeys locality principles in perturbationexpansion. But only on-shell wave functions can be defined. In the S.M., local gauge invariance may point towards the existence of equivalence classes (see later in lecture)The math of QM appears to allow for a deterministic interpretation, if only some “small, technical” problemscould be overcome.

A few of my slides shown at the beginning of the lecture were accidentally erased. Here is a summary:

2

3

1. Any live cell with fewer than two neighbours dies, as if by loneliness.

2. Any live cell with more than three neighbours dies, as if by overcrowding.

3. Any live cell with two or three neighbours lives, unchanged, to the next generation.

4. Any dead cell with exactly three neighbours comes to life.

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The use of Hilbert Space Techniques as technical

devices for the treatment of the statistics of chaos ...

, ... , , ..., , ..., , anything ... x p i

í ýA “state” of the universe:

A simple model universe: í 1ý í 2ý í 3ý í 1ý

Diagonalize:

0 0 1

1 0 0

0 1 0

U2 2 2

1 2 3, , P P P

;321

iH

i

i e

e

eU

3/2

3/2

1

23

23

0

5

ˆbut 0H ??

Emergent quantum mechanics in a deterministic system

†ˆ ˆ 2 Im( ) 0H H i f g

( ) ( )d

x t f xdt

ˆ

ˆˆ ( ) ( )

p ix

H p f x g x

ˆ( ) ( ) , ( )

dx t i x t H f x

dt

ThePOSITIVITY

Problem

5a

In any periodic system, the Hamiltoniancan be written as

2;H p

T

21 ; 0, 1, 2, ...iHT n

e H n nT

0

54321

-1-2-3-4-5

This is the spectrum of aharmonic oscillator !!

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Lock-in mechanism

In search for a

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8

Lock-in mechanism

A key ingredient for an ontological theory: Information loss

Introduce equivalence classes

{1},{4} {2} {3}

9

With (virtual) black holes, information loss will be very large! →

Large equivalence classes !

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bras

LL1LL2LL3

ketsConsider a periodic variable:

; [0, 2 ]

d

dt

, 0, 1, 2,

zH p i L

m m

0123

The quantum harmonic oscillator hasonly:

, 0, 1, 2,H n m

Beable

Changeable

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Interactions can take place in two ways.Consider two (or more) periodic variables.1:

int( );ii i i i i

dqq f q H f p

dt

Do perturbation theory in the usual way bycomputing .intn H m

This and the nect 2 slides were not shown during the lecture but may be instructive:

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1 1 12 2 2 ( 1)1; ; ;x xi i i

x x xe i e i e

intH

1q

2q2: Write for the

hopping operator.

0 1

1 0x

1 1 1(0)q q t

1 1 2 2 1 2( 1) ( ) ( )xH p p A q a q

Use

to derive 2

12

112

2

1

1exp exp ( )( 1)

if( ) 1

( )t

xt

x

i H dt i A q

A

q

a

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But this leads to decoherence !

However, in both cases,

will take values over the entire range of valuesfor n and m .

intn H m

Positive and negative values for n and mare mixed !

→ negative energy states cannot be projected out !

intn H m

But it can “nearly” be done! suppose we take many slits, and average: Then we can choose to have the desired Fourier coefficientsfor

2 2( ) ( )q f q

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Consider two non - interacting systems:

1E

2E 1 2E E

1 1 2 2H n n 15

The allowed states have “kets ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 0H E E n n n

and “bras ” with

1 11 2 1 1 2 2 1,22 2( ) ( ) , 1H E E n n n

1 2 1 2 1 20 | | | |E E E E E E

12E t

11 1 2 2 1 2 1 2 1 2 1 22 ( )( ) ( )( )E t E t E E t t E E t t

Now, and

So we also have: 1 2 1 2( ) ( )t t t t 16

The combined system is expected again to behave asa periodic unit, so, its energy spectrum must be some combination of series of integers:

1 2E E

0

5

4

3

2

1

-1

-2

-3

-4

-5

11 1 2 22( )nE n p p

For everythat are odd andrelative primes, wehave a series:

1 2,p p

17

11 1 2 22( )nE n p p

11

2

T

22

2

T

1 25, 3p p The case

121 1 2 2

2T

p p

This is the periodicity of the equivalence class:

1 1 2 2 nst odC (M 2 )p x p x

1x

2x

22

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A key ingredient for an ontological theory: Information loss

Introduce equivalence classes

í 1ý,í 4ý í 2ý í 3ý19

With (virtual) black holes, information loss will be very large! →

Large equivalence classes !20

1x

2x

1

2

This is one equivalence class

Note that is nowequivalent with

1 1 od2 ( / ) (M 2 )x k p

1x

and are beables(ontological quantities), to be determined by interactions with other variables.

1p 2p

Therefore, we should work with and instead of and1 1 1/y x p 2 2 2/y x p 1x 2x

In terms of these new variables, we have 1 2 1p p

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Some important conclusions:

An isolated system will have equivalence classes that show periodic motion; the period ω will be a “beable”.

The Hamiltonian will be H = (n +½) ω , where n is a changeable:it generates the evolution.However, in combination with other systems, n plays the role ofidentifying how many of the states are catapulted into one equivalence class: 1 1 2 /(2 1)x x n If all states are assumed to be non-equivalent, then n = 0 .The Hamiltonian for the combined system 1 and 2 is then

1212 12 1 2( ) ( )H n

The ω are all beables, so there is no problem keeping them all positive. The only changeable in the Hamiltonian is the number n for the entire universe, which may be positive (kets)or negative (bras)

This is how the Hamiltonian “locks in” with a beable !Positivity problem solved. 2

2

1. In the energy eigenstates, the equivalence classes coincidewith the points of constant phaseof the wave function.

Limit cycles

We now turn this around. Assume that our deterministic system will eventually end in a limit cyclewith period . Then is the energy (already long before the limit cycle was reached).

2 /TT

If is indeed the period of the equivalence class, then this must also be the period of the limit cycle of the system !

1 1 2 22 /( )T p p

The phase of the wave function tells us where in the limit cycle we will be.

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12( ) iH n

Since only the overall n variable is a changeable, whereas therest of the Hamiltonian, , are beables, our theory willallow to couple the Hamiltonian to gravity such that thegravitational field is a beable.

i

(Note, however, that momentum is still a changeable, and it couples to gravity at higher orders in 1/c )

Gauge theories

Gravity

The equivalence classes have so much in common with thegauge orbits in a local gauge theory, that one might suspectthese actually to be the same, in many cases ( → Future speculation)

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General conclusions

At the Planck scale, Quantum Mechanics is not wrong, but its interpretation may have to be revised, not only for philosophical reasons, but to enable us to construct more concise theories, recovering e.g. locality (which appears to have been lost in string theory).

The “random numbers”, inherent in the usual statistical interpretation of the wave functions, may well find their origins at the Planck scale, so that, there, we have an ontological (deterministic) mechanics

For this to work, this deterministic system must feature information loss at a vast scale.

We still have a problem: how to realize the information loss process before the interactions are switched on ! 2

5

Claim: Quantum mechanics is exact, yet the Classical World exists !!

The equations of motion of the (classical and [sub-]atomic) world are deterministic, but involve Planck-scale variables.

The conventional classical e.o.m. are (of course)inaccurate.

Quantum mechanics as we know it, only refers to the low-energy domain, where the fast-oscillating Planckian variables have been averaged over.

Atoms and electrons are not “real” 26

This and the next slide were not shown during lecture

At the Planck scale, we presume that alllaws of physics, describing the evolution.refer to beables only.

Going from the Planck scale to the atomicscale, we mix the bables with changeables;beables and changeables become indistinguisable !!

MACROSCOPIC variables, such as thepositions of planets, are probably againbeables. They are ontological !

Atomic observables are not, in general, beables.

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The End

G. ’t HooftA mathematical theory for

deterministic quantum mechanics

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